Can the derivative of an integral of a Brownian motion be calculated?

Question

For an equation given below: $$ \int_0^t\phi(s)dW(s) $$ where $W(s)$ is a Brownian Motion. Can I calculate the derivate w.r.t $t$ as follows: $$ \frac{d}{dt} \left( \int_0^t\phi(s)dW(s)\right) = (\phi(t)-\phi(0))(W(t)-W(0)) $$ ? Or is this wrong?

Answer 1

It is wrong. The derivative does not exist. If $\phi$ is a locally square-integrable deterministic function, the process $\int_0^\cdot \phi(s) dW_s$ has the same distribution as the process $W_{\int_0^\cdot \phi^2(s) ds}$. It is not differentiable in general since $W$ is not dirrerentiable.